Asymptotic properties of adaptive designs

There exist multiple regression applications in engineering, industry and medicine where the outcomes follow an adaptive experimental design in which the next measurement depends on the previous observations, so that the observations are not conditionally independent given the covariates. In this paper we present new asymptotic theory for such processes of dependent regression data. This includes stand-alone results for adaptive covariate processes, which, in particular, enable us to develop new likelihood theory for inference on the regression model parameters. Existing results asserting asymptotic normality of the maximum likelihood estimator require regularity conditions involving the second or third derivatives of the log-likelihood. Here we instead extend the theory of differentiability in quadratic mean (DQM) to the setting of adaptive designs, which requires strictly fewer regularity assumptions than the classical theory. In doing so, we discover a new DQM assumption, which we call summable differentiability in quadratic mean (S-DQM). As applications, we first verify asymptotic normality for two classical adaptive designs, namely the Bruceton ‘up-and-down’ design and the Robbins–Monro design. Next, we consider a more complicated problem, namely a Markovian version of the Langlie design.